Hi,
the upwind scheme is one of the simplest and most stable discretization scheme, however, it is more dissipative according to the flow you are trying to simulate.
Basically, it uses the values upstream to evaluate the property on the boundaries of the cell and then use them to compute the value at the center of the cell. As it is a "upstream" value, it takes into account the flow direction.
Also the main difference between the first and second order is basically the number of points used for the computation, i.e. one upstream point for the first order and two for the second order. Usually, it is better to use the second order scheme.
If you need more details on this, the following book is, in my opinion, very good to introduce this type of notion. Other discretization schemes are also explained.
http://www.pearson.ch/HigherEducation/PrenticeHall/1471/9780131274983/An-Introduction-to-Computational.aspx
Hi,
the upwind scheme is one of the simplest and most stable discretization scheme, however, it is more dissipative according to the flow you are trying to simulate.
Basically, it uses the values upstream to evaluate the property on the boundaries of the cell and then use them to compute the value at the center of the cell. As it is a "upstream" value, it takes into account the flow direction.
Also the main difference between the first and second order is basically the number of points used for the computation, i.e. one upstream point for the first order and two for the second order. Usually, it is better to use the second order scheme.
If you need more details on this, the following book is, in my opinion, very good to introduce this type of notion. Other discretization schemes are also explained.
http://www.pearson.ch/HigherEducation/PrenticeHall/1471/9780131274983/An-Introduction-to-Computational.aspx
Generally in short:
First Order Upwind : Easy to converge and less accurate
Second Order Upwind : More difficult to converge and more accurate than First Order
Just a suggestion:
As the second order and higher order schemes are more difficult to converge, it is better to perform a mesh convergence with the first order scheme and adjudge the need for a higher order scheme.
You can run your solver until reach some convergence, using a first order approach. After this you can chance to a second order scheme. If your mesh converges in a first order scheme, it will not necessarily be true under a second order scheme.
Hi Paulo,
I have a query regarding your suggestion. If suppose the solution obtained using first order scheme fulfils the mesh convergence criterion, and does not converge for the second order scheme, does it still mean that the solution obtained by the first order scheme is inaccurate?
And also, does this kind of a thing usually happens for complex problems?
Thanks
I also appreciate the answers above. Especially,
First Order Upwind : Easy to converge and less accurate
Second Order Upwind : More difficult to converge and more accurate than First Order
Dear Prashant Agrawal,
If your mesh converges it merely means that tour result is mesh independent, and it can still be scheme dependent. In this sense a first order approach can converge to a false result.
IMHO, an experimental validation would be the best way. This approach would clarify if a first order scheme is enough. If not, a second order scheme can be tried.
Central scheme – excessive damping and shock discontinuity are not solved. To compensate First order upwind is used. Here overshoots and undershoots is a problem -- accuracy less compare than Second order upwind scheme.
In addition to convergence and accuracy, numerical diffusion can be a major problem with first order upwind schemes and may produce False results, particularly mass diffusion problems with high Peclet numbers. Simulations using first or higher order schemes can highlight the difference, if any. May be, the local mesh Peclet number can be used to switch between the first and higher order schemes.
Diffusion problem arise the peclet and reynolds number involved for calculation of jacobian in the momentum and energy equation. so go for second order / higher order schemes.
Hi Paulo,
You suggesting to run the case using 1st order approach ..then when it converge it can be changed to second order scheme.. I have a question regarding to this matter..It might be a silly question..but may I know how can I set up for the change? because I can't see any option in solution methods for changing the order after few iteration
FIRST-ORDER UPWIND SCHEME :
Less accurate ( accuracy is of first order)as compared to second order. It assumes the face values are identical to the cell values.
SECOND-ORDER UPWIND SCHEME :
More accurate (accuracy is of second order). Taylor series expansion is used to evaluate the values at the cell face from the solution obtained at the cell centroid.
the first-order scheme has high dissipation than the second-order scheme, but the second-order scheme has more computational cost.
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